A FASTER ALGORITHM FOR CALCULATING THE INVERSE KIN...
14th
World Congress ofTFAC
Copyright
to
1999 IFAC
B-ld-lO.,S
14th Triennial World Congress, Beijing, P.R.
China
A
FASTER
ALGORITHM
FOR
CALCULATING
mE
INVERSE
KINEMATICS
OF
A
GENERAL
6R
MANIPULATOR
FOR
ROBOT
REAL
TIME
CONTROL.
Sebastian
R.
G6mez
*,
Jose
A.
Cerrada
*,
*
DIEEC
ETSII
UNED
Ciudad
Universitaria
sn
28040
Madrid, Spain.
(sgomez,
jcerrada
@ ieec.uned.es)
**
ETS
INDUSTRIALES
Campus
Universitario sn 13071 Ciudad Real, Spain
(vfeliu@ind-cr.uclm.es)
Abstract:
This
article
presents
a
new
faster
algorithm
for
the
real
time
calculation
of
the
inverse kinematics
of
general,
6R,
six
revolute
jointed,
manipulators.
It
solves
the
inverse
kinematics
problem
towards a
set
of
non-
linear
algebraic
equations
that
are
linearized
obtaining
a 16
u
,
degree
polynomial
in the
tangent
of
the half-angle
of
a revolute
joint.
Elimination
methods
were
originally presented by
Lee
and
Liang.
The
same
method
was also
derived
by
Raghavan
and
Roth,
although
the first solution with
real
time
performance
was presented
by
Manocha
and
Canny.
The
method
proposed
here
is faster
than
the
one
from
the last authors and also
avoids
the
singularities
that
could
appear
in theirs.
The
aim
of
this algorithm
is
to
find
one
of
the 16 possible solutions
in
order
to control a general
6R
manipulator
in real time
and
perform
path
following.
Using
Rhaghavan
and
Roth
method,
the
coefficients
of
the
16-degree
polynomial
are
calculated
in real
time
by
evaluating
the symbolic
determinant
that
generates
this
polynomial
in 17 different points.
The
final solution is
reached
by
a
root
solver,
beginning
the
search
in
the
last
instant
root.
The
average
performance
time
of
the
algorithm
is 25 ms in a
PC
Pentium
at
66
MHz.
This
allows
real
time
generation
of
the
references
for
the
local controllers
of
the actuators
of
the
six
joints
that
can
be
included
in
industrial manipulators. Copyright ©
1999IFAC
Keywords: Inverse Kinematics,
Efficient
Algorithms,
Real
Time
Control,
Robot
Control,
Robot
Modeling.
Copyright
©19991FAC
1.
INTRODUCTION
The
inverse
kinematics
problem
is
fundamental
in
the
design
and
control
of
robot
manipulators.
The
trajectories
of
robots
are
usually
specified
in terms
of
the
end
-
effect
pose,
position
and orientation, in
Cartesian
space, while its
motion
control
is specified
in
terms
of
joint
angles
in
joint
space.
The
inverse
kinematics
gives
the
required
joint
angles
to
reach
one
desired
pose
with the
end
- effector.
If
a real
time
control
is required, in real
time
path
following,
for
instance,
the
inverse
kinematics
must
generate
joint
angle references within
the
control
period,
which
implies
a
very
low
time
execution
for the
inverse kinematics algorithm.
Such
control
period
is
of
the
order
of
0.01 seconds.
Then
a critical
issue
in
robot
control
is
to
design
algorithms
to
carry
out
the
inverse
kinematics·
in
a
time
of
that
order
of
magnitude.
The
more
joints
the
robot
has,
the bigger
is the
complexity
of
the inverse kinematics.
The
most
interesting case
has
been
that
of
serial
manipulators
with six
degrees
of
freedom,
(DOF).
Copyright 1999 IFAC
With 6
DOF,
any
pose
in
the
space
can
be
reached.
The
complexity
of
inverse
kinematics
in
a general 6
DOF
manipulator
is a function
of
its
geometry.
This
complexity
can
be
avoided
by
'adopting
a
"closed
form"
geometry
that
is
when
three
consecutive
axes
intersect in a
common
point
or
they are parallel.
The
problems
arise
in
the general case,
when
no
such
restrictions
are
applied
to
the
manipulator.
In
fact
most
of
6-DOF
industrial manipulators
use
"closed
form"
configurations
to
avoid
calculating the inverse
kinematics
of
a general geometry.
There
are
robot
applications
where
the general
robot
geometry
is
more
efficient
than
the
before
"closed
forms".
For
example,
when
slave teleoperators
are
designed.
The
inverse
kinematics
problem
of
general
configurations has
been
studied
during
the
last three
decades, particularly
for
6
DOF
manipulators.
The
first
work
was
that
of
Pieper
(1968),
where
he
presented
the
first
inverse
kinematics
solution
based
on
iterative
numerical
techniques.
He
also
presented
the closed
form
solution.t'or a special geometry.
ISBN:
008
0432484
833